Sullivan Algebra and Trigonometry: Section 12.6 Objectives of this Section Decompose P/Q, Where Q Has Only Nonrepeated Factors Decompose P/Q, Where Q Has Repeated Factors Decompose P/Q, where Q Has Only Nonrepeated Irreducible Quadratic Factors Decompose P/Q, where Q Has Only Repeated Irreducible Quadratic Factors

Recall the following problem from Chapter R: In this section, the process will be reversed, decomposing rational expressions into partial fractions.

A rational expression P / Q is called proper if the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator.

CASE 1: Q has only nonrepeated linear factors. Under the assumption that Q has only nonrepeated linear factors, the polynomial Q has the form where none of the numbers a i are equal. In this case, the partial fraction decomposition of P / Q is of the form where the numbers A i are to be determined.

Solve this system using either substitution or elimination.

CASE 2: Q has repeated linear factors. If the polynomial Q has a repeated factor, say, (x - a) n, n > 2 an integer, then, in the partial fraction decomposition of P / Q, we allow for the terms

Solve the matrix system using Cramer’s Rule

CASE 3: Q contains a nonrepeated irreducible quadratic factor. If Q contains a nonrepeated irreducible quadratic factor of the form ax 2 + bx + c, then, in the partial fraction decomposition of P / Q, allow for the term where the numbers A and B are to be determined.

Let x = 1: -4 = 4B B = -1 Let x = -1: -12 = -4A A = 3

CASE 4: Q contains repeated irreducible quadratic factors. If the polynomial Q contains a repeated irreducible quadratic factor (ax 2 + bx + c) n, n > 2, n an integer, then, in the partial fraction decomposition of P / Q, allow for the terms where the numbers A i,B i are to be determined.